Monte Carlo 1

[ggeeoo]

Has anyone tried to use monte Carlo in structural applications?

Am trying to use monte Carlo methods to comparing vibration test results to the analysis, but not truly succeeded.

By this I mean I could not arrive at the parameters quantitatively.

Can anyone help.

 

[Enable]

I'm not a vibration expert so you'll have to provide a bit more information as to how you're using monte carlo in this context. Is there a stochastic component or optimization required in the vibration equations you're using?

If you spell out what you're trying to do I can probably help with the coding of a monte carlo simulation.

 

[ggeeoo]

Its like this .... I generate stiffness and mass matrices for a structure using Finite Element Software, or analytically.

Solve for eigen values and eigen vectors, thats what vibration equation is about.

Now, I have a set of input and output. Input is derived and is in the form of two matrices, Stiffness Matrix K and Mass Matrix M,
I have analytically obtained the outputs, Eigen values (Frequencies) and Eigen vectors (Responses). Say Estimated, Evalues and Evectors

I do the test on the structure, and i have Tested values - Tvalues and Tvectors.

There is difference between Evalues and Tvalues, Similarly, Evectors and Tvectors. And I dont know which of the input parameter caused this.

I need to use the Monte Carlo Simulation to bridge the difference and arrive at the new Input parameters.

Need help for doing monte carlo in matrices to start with.

 

[3DDave]

That sounds more like a sensitivity analysis. Bump each stiffness by some increment and see what the deltas are for all the results and, by division, get the sensitivity. Repeat for each stiffness element and see what amount each would have to change to move to the test matrix results.

Monte Carlo would be useful to determine the overall expected space of responses rather than back-driving from test data to understand a particular solution.

 

[GregLocock]

You are talking about correlating frequencies and responses, test to FEA. This is commonly done in Modal Analysis. Mostly it is done by engineering judgement, there are automated approaches using Modal Assurance Criterion but I never found them much use.

MC is at it sounds a scattergun technique to run an experiment, I can't really see how you'd use it with FEA. genetic algorithms are somewhat related and do work.

Cheers

Greg Locock


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[ggeeoo]

If we have a standard yet a sophisticated structure, for example a satellite structure, which has a number of substructures with composite materials, the responses depend on many input factors like the property of the materials, interconnections, etc. In addition to the major uncertainty, which is damping.

We use a constant structural damping in our analysis, so mist of the times, the test response (FRF) does not match with the estimated response.

Model uncertainty is definitely present in such complicated structures. So, monte Carlo can be used for quantifying the uncertainty of geometry, material, manufacturing differences, assembling and most importantly damping differences.

Hence looking for using Monte Carlo methods for solving stiffness and mass matrices.

 

[human909]

This all sounds like an academic solution in search of a problem.

 

[271828]

I'm trying to imagine what one would get from a Monte Carlo approach for this. I could imagine running through all of the combinations and not finding one that matches well, or finding a whole bunch of combinations that provide "OK" matches. I've done exercises that are fairly similar (although not comprehensive MC type analyses) and that's what happened. If this exercise would take a long time, I'd be hesitant to try it.

Instead, can you break the test specimen into smaller components or assemblies and see if the FEA matches those? Then build up toward the full structure?

 

[Enable]

In what follows I'm assuming you are setting up your equation for K*u = lambda * M * u like shown here. I'm also assuming that you what you're trying to do is use the test derived u / lambda values and back calculate acceptable K / M matrix entries.

If that's the case then the procedure below would kinda be what you would do with a few caveats. The first being that the solutions are not going to be unique, so you'll have an infinite number of acceptable combinations of entries that will produce the experimental results within epsilon. Some may be acceptable in a structural sense and others may not (unless you define the distributions to exclude these impossible combinations - see below). The second being that you'll definitely want to encode information that you know about the setup (e.g. impossible values or impossible combination of values) into your prior distributions for the various parameters - this can be difficult if we're assuming they covary as you need to define the full joint distribution.

Note: if you want to simulate effect of inputs on the matrix entries you can do that as well. For example, if entry a in Matrix M has a ~ N(u_a, var_a) then instead of assigning u_a directly you can assume u_a has its own distribution (e.g. u_a ~ N(beta_0 + beta_1 * input_1 + beta_2*input_2 + ..., var_input 1 + var_input 2 + ...)

On the other hand, if you have ample computing power you are free to just simulate from independent distributions, record those who produce results within epsilon and compare to your analytically derived results to see which of your simulations is closest. This is what the below protocol pretty much does (step 6 not shown is store and compare to your analytical results). You're likely to get lost in the high-dimensional space here though so defining the joint distribution and using MCMC is definitely the way to go IMHO.

As a note on efficiency I would probably avoid doing any inversions (computationally expensive) and would arrange my simulations / math accordingly. So my step 5 isn't how I would actually implement the thing but that's the concept.

Another note on efficiency is that if you can bound the various matrix entries, and establish incremental changes of interest, then you can randomly sample from the vector containing the bounds and all entries in-between. This will produce a large number of combinations, but infinitely less than assuming continuous distributions as I did in the procedure below.

 

[271828]

As mentioned above, the Modal Assurance Criterion is used for situations like this.

The issue is you might need to assign parameters outside the reasonable range to improve the MAC value and still might not end up with a very good match. This is probably not one of those "I found the answer!" situations. LOL

 

[GregLocock]

I don't see how damping will affect MAC, as the mode shapes won't change. So, you really need to be comparing response spectra (or time histories), test to FEA. Developing a method to compare that is tricky. Luckily in my case other people have decided how to encapsulate a spectrum or time history as a few numbers.

I usually run single factor experiments to see what I have to poke and then maybe a DOE. You can use MC to populate a DOE but is not computationally efficient, you end up with a lot of runs clustered around the centre that don't help much. The advantage of MC over a series of fixed levels is that if a particular level of a factor causes the run to fail, then it'll only affect a few of your MC runs, instead of wiping out a whole row. Also you don't have to be a mathematician to add more runs to investigate particular factor.

Cheers

Greg Locock


New here? Try reading these, they might help FAQ731-376

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[ggeeoo]

Thankyou very much to all.

My special thanks to Enable for taking out the time to explain it clearly. I had tried till step 3. Yet to use the prior information into the distributions (Bayesian). It's taking time to apply mcmc and Gibbs though the concept is understood.

Will try and come back. Thankyou.